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I feel frustrated that we cannot conceive of numbers like 10^80 (atoms in the universe) or 10^4000 (number configurations for a system with 4000 variables havin

by hughw 2y ago

I feel frustrated that we cannot conceive of numbers like 10^80 (atoms in the universe) or 10^4000 (number configurations for a system with 4000 variables having 10 states each). Maybe there are superbrains out there in the universe that can do so.

crazygringo 2y ago
I guess you have to define what you mean by "conceive". I'm not sure you can even conceive a number like 1,000, if you're talking about holding an intuitive visual understanding in your mind at once. Like, I can easily see 100 in my mind's eye as a 10x10 grid of circles. Even if I don't see each one clearly, I have a good sense of the 10 on each edge and the way it fills in. But ask me to imagine 10 of those side-by-side to make 1,000, and I don't think I can. Once I imagine the 10 groups, each one is just a square simplification, rather than any individual pieces within. But I'm totally familiar with 1,000 as a concept I can multiply and divide with, and I can do math with 10^80 as well. And I can do so fairly "intuitively" as well -- it's just all the numbers up to 80 digits long. Even 4,000 digits fits on a single page of a book.
- hughw 2y ago
  My first cut at conceiving is to answer "how long would it take a really fast computer to count to that number". The answer for 10^4000 is still something like 10^3978 years. So, a still inconceivable time. (100 tera-ops computer) [edited to correct calculation)
  - crazygringo 2y ago
    But the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric, no? Your idea arbitrarily selects numbers around 2 billion as being conceivable, at least for a single core on my MacBook. But my question isn't what makes 10^4000 inconceivable -- my question is what makes 10^4000 any less conceivable than 1000. To me, they're both firmly in the realm of abstractions we can reason about using the same types of mathematical methods. They're both qualitatively different from numbers like 5 or 10 which are undoubtedly "conceivable".
    - hughw 2y ago
      the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric But I'd just call it "instantaneous" and something I experience frequently. Whereas 10^3978 years is beyond experience and even imagination.
      - crazygringo 2y ago
        > But I'd just call it "instantaneous" and something I experience frequently. Then since you're OK with all smaller numbers being equally "instantaneous", then 1/10^4000 seconds is instantaneous too. Add up enough of those to make a second, and you can conceive of your previously inconceivable number! :) Of course, that will seem silly. I'm just illustrating I don't think there are any grounds for claiming exponentially large numbers are inconceivable, but exponentially small numbers are somehow conceivable. They're just as far from 1, multiplicatively, no matter which direction you go in.
        hughw 2y ago
        All right! I cannot count to any fractional number, even conceivable ones, so using the counting time to assess conceivability fails for all fractions. For small numbers, groping for any experience in the macro world fails once you get below the Planck scale. I'm afraid you might have broken the fabric of the universe by even typing such a small number.
MisterTea 2y ago
To me it's hard because we don't come across these magnitudes on a daily basis. Grains of sand on a beach is an off the top of my head way to visualize such quantities. But yeah, numbers that big, are so big that we have nothing to compare them with.
kqr 2y ago
I feel like I have a much better grasp of those numbers since I started using logarithms for mental maths: https://entropicthoughts.com/learning-some-logarithms https://entropicthoughts.com/learning-some-logarithms
avmich 2y ago
Read http://mrob.com/pub/math/largenum.html http://mrob.com/pub/math/largenum.html . After that you'll probably look easier to those numbers.

by hughw 2y ago

crazygringo 2y ago

I guess you have to define what you mean by "conceive". I'm not sure you can even conceive a number like 1,000, if you're talking about holding an intuitive visual understanding in your mind at once. Like, I can easily see 100 in my mind's eye as a 10x10 grid of circles. Even if I don't see each one clearly, I have a good sense of the 10 on each edge and the way it fills in. But ask me to imagine 10 of those side-by-side to make 1,000, and I don't think I can. Once I imagine the 10 groups, each one is just a square simplification, rather than any individual pieces within. But I'm totally familiar with 1,000 as a concept I can multiply and divide with, and I can do math with 10^80 as well. And I can do so fairly "intuitively" as well -- it's just all the numbers up to 80 digits long. Even 4,000 digits fits on a single page of a book.

hughw 2y ago
My first cut at conceiving is to answer "how long would it take a really fast computer to count to that number". The answer for 10^4000 is still something like 10^3978 years. So, a still inconceivable time. (100 tera-ops computer) [edited to correct calculation)
- crazygringo 2y ago
  But the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric, no? Your idea arbitrarily selects numbers around 2 billion as being conceivable, at least for a single core on my MacBook. But my question isn't what makes 10^4000 inconceivable -- my question is what makes 10^4000 any less conceivable than 1000. To me, they're both firmly in the realm of abstractions we can reason about using the same types of mathematical methods. They're both qualitatively different from numbers like 5 or 10 which are undoubtedly "conceivable".
  - hughw 2y ago
    the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric But I'd just call it "instantaneous" and something I experience frequently. Whereas 10^3978 years is beyond experience and even imagination.
    - crazygringo 2y ago
      > But I'd just call it "instantaneous" and something I experience frequently. Then since you're OK with all smaller numbers being equally "instantaneous", then 1/10^4000 seconds is instantaneous too. Add up enough of those to make a second, and you can conceive of your previously inconceivable number! :) Of course, that will seem silly. I'm just illustrating I don't think there are any grounds for claiming exponentially large numbers are inconceivable, but exponentially small numbers are somehow conceivable. They're just as far from 1, multiplicatively, no matter which direction you go in.
      - hughw 2y ago
        All right! I cannot count to any fractional number, even conceivable ones, so using the counting time to assess conceivability fails for all fractions. For small numbers, groping for any experience in the macro world fails once you get below the Planck scale. I'm afraid you might have broken the fabric of the universe by even typing such a small number.

MisterTea 2y ago

To me it's hard because we don't come across these magnitudes on a daily basis. Grains of sand on a beach is an off the top of my head way to visualize such quantities. But yeah, numbers that big, are so big that we have nothing to compare them with.

kqr 2y ago

I feel like I have a much better grasp of those numbers since I started using logarithms for mental maths: https://entropicthoughts.com/learning-some-logarithms https://entropicthoughts.com/learning-some-logarithms

avmich 2y ago

Read http://mrob.com/pub/math/largenum.html http://mrob.com/pub/math/largenum.html . After that you'll probably look easier to those numbers.